where the constant \(c_{np}\)  depends only on \(n\)  and \(p\) . Here \(f(X)\)  is the multiplication by \(f\)  in position space and \(g(-i\nabla )\)  is the multiplication by \(g\)  in momentum space. We refer to Section  for background on Schatten classes and weak Schatten classes.
Cwikel's estimates were extended to \(p\in (0,2)\)  by Birman-Solomyak [1]} and to \(p=2\)  by Birman-Karadzhov-Solomyak [2]} (who also dealt with more general Lorentz ideals \({L}_{p,q}\) ). Solomyak [3]} substantially improved the understanding of the \(p=2\)  case in even dimension.
